Table 3.

Outer and inner leading order dynamics of the ST–PDE and P–PDE for all positive values of $\beta$

	Leading order stalk cell dynamics	Leading order tip cell dynamics $(y=X-\frac{\tau +X_0}{ϵ},X_0\in \mathbb{R})$
$\mathit{\beta }\sim O(1)$	$\frac{\partial w_0}{\partial \tau }=\beta u_0$	$\frac{\partial U_0}{\partial \tau }=\frac{{\partial }^2{U}_0}{\partial y^2}+U_0(\tilde{C}-U_0-W_0)$
$\mathit{\beta }\gg \mathbf{1}$	$\frac{\partial w_0}{\partial \tau }=\tilde{\beta }{\overline{u}}_0$	$\frac{\partial {\overline{U}}_0}{\partial \tau }=\frac{{\partial }^2{\overline{U}}_0}{\partial y^2}+{\overline{U}}_0(\tilde{C}-W_0)$
($\tilde{\beta }=\beta {ϵ}^m$, $m>0$)
$\mathit{\beta }\ll \mathbf{1}$	$\frac{\partial {\overline{w}}_0}{\partial \tau }=\widehat{\beta }{u}_0$	$\frac{\partial U_0}{\partial \tau }=\frac{{\partial }^2{U}_0}{\partial y^2}+U_0(\tilde{C}-U_0)$
($\widehat{\beta }=\beta {ϵ}^{-m}$, $m>0$)

In all cases, $u(X,\tau )$ and $U(y,\tau )$ represent the tip cell density, $w(X,\tau )$ and $W(y,\tau )$ the stalk cell density, and $c(X,\tau )$ and $\tilde{C}(y,\tau )$ the TAF concentration. We refer to Sect. 2 and Appendices A and B for the derivation of these systems